A298321 The Nekrasov-Okounkov sequence.

1, 1, 1, 1, 2, 3, 3, 4, 3, 8, 6, 9, 8, 9, 12, 13, 11, 13, 12, 16, 18, 19, 18, 19, 21, 22, 22, 24, 24, 27, 25, 26, 29, 28, 31, 33, 32, 34, 32, 37, 35, 36, 37, 38, 42, 42, 41, 42, 43, 46, 48, 48, 45, 48, 50, 53, 54, 54, 51, 56, 56, 55, 58, 59, 60, 62, 62, 62

Offset: 1

Author

Kenta Suzuki, Jan 17 2018

Keywords

• nonn

Comments

a(n) is the degree in terms of z of the coefficient of x^n's highest degree irreducible factor in Product_{m>=1} (1-x^m)^(z-1). This can be calculated by reducing the polynomial in the Nekrasov-Okounkov formula.

Examples

			For n = 5, a(n) = 2 because the coefficient of x^5 is Product_{m>=1} (1-x^m)^(z-1). This can be factorized as -(z-7)*(z-4)*(z-1)*(z^2 -23*z + 30)/120.
		

Links

• Vincent Delecroix, Table of n, a(n) for n = 1..506
• Vincent Delecroix, SageMath script to generate the sequence
• Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, Annales de l'institut Fourier, 60 no. 1 (2010), pp. 1-29.
• Nikita A. Nekrasov and Andrei Okounkov, Seiberg-Witten Theory and Random Partitions, arXiv:hep-th/0306238, 2003.

Crossrefs

Cf. A210590, A234937.

Programs

Julia

using Nemo
function A298321(len)
    R, z = PolynomialRing(ZZ, 'z')
    Q = [R(1)]; S = [1, 1, 1, 1]
    for n in 1:len-4
        p = z*sum(sigma(ZZ(k), 1)*risingfac(n-k+1, k-1)*Q[n-k+1] for k in 1:n)
        push!(Q, p)
        for (f, m) in factor(p)
            deg = degree(f)
            deg > 1 && push!(S, deg)
        end
    end
S end
A298321(72) |> println # Peter Luschny, Oct 27 2018, after Vincent Delecroix

Mathematica

(* This naive program is not suitable to compute a large number of terms *) a[n_] := a[n] = SeriesCoefficient[Product[(1-x^m)^(z-1), {m, 1, n}], {x, 0, n}] // Factor // Last // Exponent[#, z]&;
Table[Print[n, " ", a[n]]; a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 18 2019 *)

PARI

{a(n) = vecmax(apply(x->poldegree(x), factor(polcoef(prod(k=1, n, (1-x^k+x*O(x^n))^(z-1)), n))[, 1]))} \\ Seiichi Manyama, Nov 07 2020

Extensions

More terms from Vincent Delecroix, Oct 05 2018